arXiv subjects

Jasem Hamoud

Publications and source records attributed to Jasem Hamoud.

At least 19 recordsLinked to original sources

arXiv2026

$\sigma$-Irregularity of Trees with Prescribed Maximum Degree: A Majorization--Duality--Stability Framework

Trees of maximum $\sigma$-irregularity subject to a prescribed maximum degree $\Delta$ have been characterized for $\Delta=4$ and $\Delta=5$, with the extension to arbitrary $\Delta\geqslant 3$. This paper develops a unified framework for this class of problems built on three pillars. A majorization-theoretic reformulation that decomposes $\sigma$-extremization into a Schur-convex optimization over degree sequences and a rearrangement type optimization over tree realizations, valid for every $\Delta\geqslant 3$. We establish a duality between the maximization and minimization problems, alongside the general $\Delta$ closed form for $\sigma_{\max}(n,\Delta)$. A stability result showing that the optimality gap between extremal and near extremal trees is bounded independently of $n$ and grows as $\Theta(\Delta^3)$.

math.GM↗

arXiv2026

Duality in Biperiodic Fibonacci Words Substitution Frequencies and Combinatorial Invariants

In this paper, a natural duality on the family of biperiodic Fibonacci words $\mathfrak{F}^{(a,b)}$ generated by the directive sequence $(a,b,a,b,\dots)$. Both of $\mathfrak{F}^{(a,b)}$ and $\mathfrak{F}^{(b,a)}$ are related by the explicit morphism $\sigma_a:0\mapsto 0^a1$, $1\mapsto 0$, establishing a precise substitutional correspondence between them. We compute the exact letter frequencies, give a complete description of the return words for each letter, prove the existence of arbitrarily long palindromic prefixes, and determine the continued fraction expansion of the slope $\theta^{(a,b)}$. These findings reveal that the apparent asymmetry in several invariants arises uniformly from the length-redistribution mechanism induced by the morphism $\sigma_a$.

math.GM↗

arXiv2026

On Epimorphisms of Hypergraphic Automata and Input Symbol Semigroups

Hypergraphic automata are automata whose state sets and output symbol sets are hypergraphs invariant under the actions of the transition and output functions. Universally attracting objects in the category of such automata are called universal hypergraphic automata; their semigroups of input symbols are algebras of mappings whose properties are tightly linked to the algebraic structure of the automata themselves. This paper establishes a complete characterisation of epimorphisms of universal hypergraphic automata and of their semigroups of input symbols. A central contribution is the introduction of two distinct notions of epimorphism for hypergraphs including weak, strong and the proof that these notions diverge in general but necessarily coincide for the important subclass of $p^*$-hypergraphs, which includes automata whose state hypergraphs and output hypergraphs are projective or affine planes. The main results give necessary and sufficient conditions for a triple $(f, \mathbb{P}_s, g)$ to be an epimorphism of universal hypergraphic automata, expressed in terms of the component maps on the state and output hypergraphs.

cs.FL↗

arXiv2026

Sharp Bounds and Extremal Fuzzy Graphs for the Fuzzy Sombor Index

The fuzzy Sombor index applies the classical Sombor index to fuzzy graphs, incorporating both edge membership values and fuzzy vertex degrees. For $\alpha>1$, the general fuzzy Sombor index it is defined as \[ \mathrm{SO}^{\mu}_{\alpha}(\Gamma)=\sum_{uv\in V(\Gamma)} \left( \mu(u,v)\, \sqrt{\mu_u^2+\mu_v^2} \right)^{\alpha}. \] This paper analyses extremal features of $\mathrm{SO}^{\mu}$ across different types of fuzzy graphs. We determine the maximum value (resp. minimum value) of $\mathrm{SO}^{\mu}$ characterise in regular fuzzy graph. We established significant inequality between the fuzzy Sombor index and other well-known fuzzy topological indices.

math.GM↗

arXiv2026

Structural Components Dominate Asymptotic Behavior on Sombor Index with Iterated Pendant Constructions

The Sombor index, a degree-based topological descriptor introduced by Gutman in 2021, lacks closed-form expressions for complex hierarchical trees with multi-level pendant structures and nonuniform degree distributions, despite extensive results for simpler families such as paths, stars, cycles, and basic caterpillars. For a simple graph $\mathcal{G}$, the Sombor index is defined as \[ \mathrm{SO}(\mathcal{G}) = \sum_{uv \in E(\mathcal{G})} \sqrt{d(v)^2 + d(u)^2}. \] In this work, we derive a general recursive formula for the Sombor index of multi-level pendant-augmented path trees. These trees are constructed from a spine path $\mathcal{P}_n$ ($n \ge 2$) in which each vertex has degree $2+k$ and are iteratively augmented over $m \ge 1$ hierarchical levels. Pendants attached to odd-indexed spine vertices branch with replication factor $k$ and terminal degree $\ell_i$, whereas those stemming from even-indexed vertices incorporate an initial offset $\ell_1>2$ that propagates through subsequent levels. These results significantly advance the theoretical and computational study of degree-based topological descriptors in iteratively constructed graphs.

math.GM↗

arXiv2026

Theta-Relations Among Degree-Based Tree Indices

In this paper, degree-based topological indices play a key role in the structural analysis of graphs in this paper and have significant uses in chemical graph theory. We investigate the connections between three such tree indices: the Albertson, Sombor, and Sigma indices. We show that the quadratic degree deviation, measured by the Sigma index, tightly controls the Sombor index of a tree by establishing sharp two-sided bounds. We demonstrate that the Sombor and Sigma indices are asymptotically equivalent up to constant factors as a direct result. A pure $\Theta$-relationship between the Sombor index and the Albertson index is derived by taking into account extremal trees with a fixed degree sequence. This finding demonstrates that, in extremal configurations, quadratic degree interactions and absolute degree disparities scale appropriately. Overall, our data suggest that the Sombor index functions as an intermediate descriptor, capturing both global degree dispersion and local edge irregularity. From a structural standpoint, these findings clarify the relationship between vertex-based and edge-based irregularity measurements in trees.

math.CO↗

arXiv2026

On Extremal Family Trees $(\mathcal{T}_n)_{n\geqslant 3}$ Beyond Caterpillars and Greedy Constructions

This paper investigates topological indices for the greedy tree $\mathcal{T}_\mathscr{D}$ associated with a graphic degree sequence $\mathscr{D} = (d_1 \geqslant d_2 \geqslant \dots \geqslant d_n)$ of a tree. A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices. We investigate the extremal properties of the graph invariant $\sigma$ over the family $\mathcal{T}_n$ of all trees on $n \ge 3$ vertices. Specifically, we compare the minimum values of $\sigma$ attained in restricted subclasses -- including caterpillar trees and greedy trees -- with the global minimum over $\mathcal{T}_n$. We prove that caterpillar trees do not achieve the minimum value of $\sigma$ among all trees, whereas greedy trees attain values no smaller than this global minimum. Moreover, we show that certain trees, which are neither caterpillars nor greedy trees, have $\sigma$-values strictly between the global minimum over $\mathcal{T}_n$ and the minimum among caterpillar trees. These results highlight structural limitations of these common tree classes in extremal problems and offer new insights into the role of non-caterpillar, non-greedy trees in minimizing graph invariants.

math.GM↗

arXiv2026

Analysis of the Density of Words under Morphism $\{a,b\}$

In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.

math.GM↗

arXiv2025

Bounds on the Albertson Index for Trees with Given Degree Sequences

In this paper, we presents novel and sharp bounds on the Albertson index of trees, revealing deep connections between degree sequences and graph irregularity where the Albertson index of Caterpillar tree satisfy \[ \operatorname{irr}(G)=\left( {{d_n} - 1} \right)^2 + \left( {d_1 - 1} \right)^2 + \sum\limits_{i = 2}^{n - 1} {\left( {{d_i} - 1} \right)\left( {{d_i} - 2} \right)} +\sum_{i=1}^{n-1}|d_i-d_{i+1}|. \] We derive powerful inequalities that precisely characterize the minimum and maximum values of the Albertson index, incorporating intricate dependencies on vertex degrees, edge counts, and the average of elements in degree sequence $\mathscr{D}=(d_1,d_2,\dots,d_n)$ where $d_n\geqslant d_{n-1}\geqslant \dots\geqslant d_2\geqslant d_1$. Our results not only improve existing extremal bounds but also uncover striking relationships between the structure of trees and their irregularity measurements. These advances open new avenues for the analysis of graph irregularity and contribute essential tools for the study of degree-based topological indices in combinatorial graph theory.

math.GM↗

arXiv2025

On the Asymptotic Palindrome Density of Fibonacci Infinite Words

In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities.

math.CO↗

arXiv2025

The Orientalists' Stance Towards Arabic Sciences (Especially Arabic Astronomy)

In this paper, we highlight the influence of Arab/Islamic civilization in the field of the history of astronomy on European historians. We also aim to elucidate the stance of Orientalists toward the study of Arab sciences and to clarify their orientations, with a particular focus on astronomy, while revealing the significant role played by Arab scholars in this domain and the impact of their contributions-especially astronomical tables (zij)-on Western astronomers. Furthermore, we have clarified the mechanisms of transmission of Arab sciences, particularly astronomy, from Arab scholars to Western scholars, and the role of Arab astronomers in Western civilization. In addition, we address the contributions of Arab scholars to the development of astronomy and the perspective of Orientalists, particularly David King, regarding this matter. We also underscore the importance of Orientalists' works in analyzing Arab/Islamic scholarly output, identifying its influence on the West in the field of astronomy, and demonstrating how Western scholars benefited from translations of Arabic books in this discipline. In this paper, we adopt the historical retrieval methodology, by referencing previously documented astronomical information and contributions, with an emphasis on the processes of transmission of these sciences from the Arabs to the West.

math.HO↗

arXiv2025

A Study of NP-Completeness and Undecidable Word Problems in Semigroups

In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$ implies the existence of a deterministic Turing machine solving $L$ in polynomial time $O(n^k)$. Central to our investigation is polynomial reducibility. Also, we demonstrate the existence of an associative calculus $A(\mathfrak{T})$ with an algorithmically undecidable word problem, where for a Turing machine $\mathfrak{T}$ computing a non-recursive function $E(x)$, we establish that $q_1 01^x v \equiv q_0 01^i v \Leftrightarrow x \in M_i$ for $i \in \{0,1\}$, where $M_i = \{x \mid E(x) = i\}$. This connection between computational complexity and algebraic undecidability illuminates the fundamental limits of algorithmic solutions in mathematics.

cs.CC↗

arXiv2025

The Effect of Using Popular Mathematical Puzzles on The Mathematical Thinking of Syrian Schoolchildren

In this paper we provide a good overview of the problems and the background of mathematics education in Syrian schools. We aimed to study the effect of using popular mathematical puzzles on the mathematical thinking of schoolchildren, by conducting a paired experimental study (pre-test and post-test control group design) of the data we obtained through a sample taken from students of sixth-grade primary school students in Syria the Lady Mary School in Syria, in order to evaluate the extent of the impact of popular mathematical puzzles on students' ability to solve problems and mathematical skills, and then the skills were measured and the results were analyzed using a t-test as a tool for statistical analysis.

econ.TH↗

arXiv2025

The Role of Mathematical Folk Puzzles in Developing mathematical Thinking and Problem-Solving Skills

This paper covers a variety of mathematical folk puzzles, including geometric (Tangrams, dissection puzzles), logic, algebraic, probability (Monty Hall Problem, Birthday Paradox), and combinatorial challenges (Eight Queens Puzzle, Tower of Hanoi). It also explores modern modifications, such as digital and gamified approaches, to improve student involvement and comprehension. Furthermore, a novel concept, the "Minimal Dissection Path Problem for Polyominoes," is introduced and proven, demonstrating that the minimum number of straight-line cuts required to dissect a polyomino of N squares into its constituent units is $\mathrm{N}-1$. This problem, along with other puzzles, offers practical classroom applications that reinforce core mathematical concepts like area, spatial reasoning, and optimization, making learning both enjoyable and effective.

econ.TH↗

arXiv2025

Closed-Form Analysis and Extremal Bounds of Albertson and Sigma Indices in Trees with Prescribed Degree Sequences

This study explores the irregularity properties of trees with prescribed degree sequences by analyzing two prominent topological indices: the Albertson index and the sigma index. With a particular emphasis on caterpillar trees -frequently used to model molecular chains- we derive a closed-form expression for the Albertson index: \[ \mathrm{irr}(\mathscr{C}(n,m)) = m(m+1)n - 2m + 2, \quad \text{for } n \geq 3. \] Furthermore, we establish extremal bounds for both indices across tree families characterized by fixed degree sequences. The results yield a unified analytical framework for comparing linear and quadratic irregularity measures, and provide new structural insights relevant to applications in chemical graph theory and extremal graph analysis.

math.CO↗

arXiv2025

Optimal Behaviour in Extremal Bounds for $\sigma$-Irregularity

In this paper, we establishe the extremal bounds of the topological indices -- Sigma index -- focusing on analyzing the sharp upper bounds and the lower bounds of the Sigma index, which is known $\sigma(G)=\sum_{uv\in E(G)}(d_G(u)-d_G(v))^2$. We establish precise lower and upper bounds for the Sigma index, leveraging a non-increasing degree sequence $\mathscr{D} = (d_1, d_2, \dots, d_n)$, A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices.

math.CO↗

arXiv2025

Extremal Bounds on the Sigma and Albertson Indices for Non-Decreasing Degree Sequences

We establish sharp extremal bounds on the Albertson and Sigma irregularity indices for trees with prescribed degree sequences, with emphasis on caterpillar trees as key extremal configurations. New lower and upper bounds are derived in terms of maximum degree, average degree, and auxiliary sequence parameters, highlighting the quadratic growth of the Sigma index relative to the linear Albertson index. Closed-form expressions, direct index relations, and empirical validation confirm the bounds' tightness. These findings extend prior work on linear irregularity measures and offer precise tools for analyzing degree-heterogeneous trees in graph theory and chemical graph applications.

math.CO↗

arXiv2025

Density Characterization with The Upper Bound of Density of Fibonacci Word

This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote the units digit. Through analysis of these ratios and their convergence to powers of $\varphi$, we illustrate the intrinsic exponential growth rates characteristic of Fibonacci words. By considering the natural density concept for sets of positive integers, it is demonstrated that the density of Fibonacci words approaches unity, correlating with classical results on Fibonacci number distributions as \[ \operatorname{DF}(F_k) <\frac{m(m+1)}{n(2m-n+1)}. \] Furthermore, generating functions and combinatorial formulas for general terms of Fibonacci words are derived, linking polynomial expressions and limit behaviors integral to their combinatorial structure. The study is supplemented by numerical data and graphical visualization, confirming theoretical findings and providing insights into the early transient and asymptotic behavior of Fibonacci word densities.

math.CO↗